Trademark rights arise through use of the mark in commerce to identify a product or service. Registering a trademark gives you the ability to enforce the mark more broadly than if you didn't register the mark, but the underlying rights arise through use in commerce. There is no requirement for registration -- you acquire rights by using the mark.
So, it is possible that someone is using the name you want as a trademark, and they have acquired trademark rights to it (known as "common law" trademarks). That doesn't mean you can't register the mark, especially if you are in a different geographic region, or in a different line of business. In that case, you would both have rights to the mark. You, as the "junior" mark user, would not have the right to stop the "senior" user from using the mark; however, if you obtain a registration on the mark, you can prevent a senior user (there can be more than one, even) from expanding the geographical scope of their business.
That's one of the advantages of registration. There are others as well, mainly revolving around how one might enforce their mark.
But a trademark is not like a patent -- you don't stake out a claim, and nobody else gets to use it. You have to acquire the rights through use of the mark (registration doesn't give you the rights, you have to acquire them), and then continue to use and enforce your mark, to retain the rights. And there may well be cases where there are others using the same name as well.
Now, if you are in the same (or close) line of business, and their common-law mark is well known, or if you are in the same geographical area (or they are national in scope), the strength of their common-law mark may be sufficient to prevent you from registering the same (or confusingly similar) mark. But if someone else just has a URL (URL's do not necessarily establish trademark rights), or is in a small geographical region separate from where you are, or is in a foreign country, then you may very well be able to obtain your own trademark rights and registration to the mark.
I realize that this is probably a bit confusing, but this is more-or-less the way it works.